Nicholson-Shain Theory for Cyclic Voltammetry

The Nicholson-Shain theory (1964) describes cyclic voltammetry for irreversible and quasi-reversible electrochemical reactions, extending the Randles-Sevcik equation by including electron transfer kinetics. The peak current for an irreversible reaction is:

\[ i_p = (2.99 \times 10^5) \, n ( \alpha n_a)^{1/2} A D^{1/2} v^{1/2} c_0 \]

where \( n \) is the number of electrons, \( \alpha \) is the transfer coefficient, \( n_a \) is the number of electrons in the rate-determining step, \( A \) is the electrode area (cm²), \( D \) is the diffusion coefficient (cm²/s), \( v \) is the scan rate (V/s), and \( c_0 \) is the bulk concentration (mol/cm³).

Step 1: Context and Fick's Second Law

In cyclic voltammetry, the potential is swept linearly (\( E = E_i \pm v t \)), and the current peaks when the potential favors the redox reaction. For an irreversible reaction (\( \text{O} + n e^- \to \text{R} \)), the current is diffusion-limited but influenced by slow electron transfer kinetics. The concentration is governed by Fick's Second Law:

\[ \frac{\partial c_O}{\partial t} = D_O \frac{\partial^2 c_O}{\partial x^2} \]

where \( c_O(x, t) \) is the concentration of the oxidized species, and \( D_O \) is its diffusion coefficient. Similar for the reduced species (\( c_R \), \( D_R \)). Assume \( D_O \approx D_R = D \).

Step 2: Kinetic Boundary Condition

Unlike reversible systems (where the Nernst equation applies), irreversible reactions have a slow electron transfer rate, described by the Butler-Volmer equation. The flux at the electrode (\( x = 0 \)) is:

\[ J_O(0, t) = -k^0 c_O(0, t) \exp\left( -\frac{\alpha n_a F}{R T} (E(t) - E^0) \right) \]

where \( k^0 \) is the standard heterogeneous rate constant, \( \alpha \) is the transfer coefficient, and \( E(t) = E_i - v t \) (reductive scan). The current is:

\[ i = n F A J_O(0, t) \]

Boundary conditions:

At \( x = 0 \): Fluxes of \( \text{O} \) and \( \text{R} \) are equal and opposite (\( J_O = -J_R \)).
At \( x \to \infty \): \( c_O(x, t) = c_0 \), \( c_R(x, t) = 0 \).
At \( t = 0 \): \( c_O(x, 0) = c_0 \), \( c_R(x, 0) = 0 \).

Step 3: Dimensionless Formulation

To solve the diffusion equation, Nicholson and Shain used dimensionless variables to simplify the time-dependent potential and kinetics. Define:

\[ \xi = x \sqrt{\frac{n F v}{R T D}}, \quad \tau = \frac{n F v t}{R T}\] \[\quad \psi = \frac{i}{n F A c_0 \sqrt{D n F v / R T}} \]

The diffusion equation becomes:

\[ \frac{\partial c_O}{\partial \tau} = \frac{\partial^2 c_O}{\partial \xi^2} \]

The boundary condition at \( x = 0 \) incorporates the kinetic term, with the rate constant scaled as:

\[ \Lambda = \frac{k^0}{\sqrt{D n F v / R T}} \]

For irreversible systems, \( \Lambda \) is small (slow kinetics), and the reaction is kinetically limited.

Step 4: Peak Current for Irreversible Systems

Nicholson and Shain solved the diffusion equation numerically, finding the peak current occurs when the surface concentration gradient is maximized. For an irreversible reaction, the dimensionless peak current is:

\[ \psi_p \approx 0.496 (\alpha n_a)^{1/2} \]

The actual current is:

\[ i_p = n F A c_0 \sqrt{\frac{n F D v}{R T}} \cdot 0.496 (\alpha n_a)^{1/2} \]

Evaluating constants (\( F = 96485 \), \( R = 8.314 \), \( T = 298 \)):

\[ \sqrt{\frac{F}{R T}} \approx 6.24 \times 10^3 \, \text{C}^{1/2} \text{mol}^{-1/2} \text{K}^{-1/2} \]

Thus:

\[ i_p = (0.496) n^{3/2} (\alpha n_a)^{1/2} F A c_0 \sqrt{\frac{D v}{R T / F}} \]

Converting units and combining constants gives:

\[ i_p = (2.99 \times 10^5) \, n ( \alpha n_a)^{1/2} A D^{1/2} v^{1/2} c_0 \]

Step 5: Quasi-Reversible Systems and Applications

For quasi-reversible systems, Nicholson introduced a kinetic parameter \( \psi \) to interpolate between reversible (\( \Lambda \gg 1 \)) and irreversible (\( \Lambda \ll 1 \)) limits. The theory also predicts shifts in peak potential and wave shape, useful for determining \( k^0 \) and \( \alpha \). Applications include characterizing electrode kinetics and diffusion coefficients in cyclic voltammetry experiments.

Realated Topics:
Derivation of Randles Sevcik Equation
Derivation of Cottrell Equation
Derivation of Ilkovic Equation

Nicholson-Shain Theory for Cyclic Voltammetry